prompt listlengths 1 1 | label stringlengths 1 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一颗质地均匀的正方体骰子，六个面上分别标有点数 $1,2,3,4,5,6$. 随机地抛掷该骰子三次 (各次抛掷结果相互独立), 所得的点数依次为 $a_{1}, a_{2}, a_{3}$, 则事件 \" $\\left|a_{1}-a_{2}\\ri... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求具有下述性质的最小正数 $c$ ：对任意整数 $n \\geq 4$,以及集合 $A \\subseteq\\{1,2, \\cdots, n\\}$, 若 $|A|>c n$, 则存在函数 $f: A \\rightarrow\\{1,-1\\}$... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若等差数列 $\\left\\{a_{n}\\right\\}$ 及正整数 $m(m \\geq 3)$ 满足: $a_{1}=1, a_{m}=2$, 且 \n$$ \n\\frac{1}{a_{1} a_{2}}+\\frac{1}{a_{2}... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 $\\left\\{a_{n}\\right\\}$ 的各项均为非负实数, 且满足：对任意整数 $n \\geq 2$, 均有 $a_{n+1}=a_{n}-a_{n-1}+n$. 若 $a_{2} a_{2022}=1$, 求 $a_{1}... | 6076 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $\\triangle A B C$ 的内角 $A, B, C$ 满足 $\\sin A=\\cos B=\\tan C$, 求 $\\cos ^{3} A+\\cos ^{2} A-\\cos A$ 的值.原始的答案是\\frac{m}{n}的形... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, 已知\n$\\cos C=\\frac{\\sin A+\\cos A}{2}=\\frac{\\sin B+\\cos B}{2}$, 求 $\\sin C$ 的值原始的答案是\\frac{m}{n}的... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\\operatorname{def}$ 是奇数的概率为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $1,2,3, \\cdots, 10$ 中随机选出一个数 $a$ ，在 $-1,-2,-3, \\cdots,-10$ 中随机选出一个数 $b$ ，则 $a^{2}+b$ 被 3 整除的概率为 $\\qquad$.原始的答案是\\frac{m}{... | 137 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $A B=6, B C=4$, 边 $A C$ 上的中线长为 $\\sqrt{10}$, 则 $\\sin ^{6} \\frac{A}{2}+\\cos ^{6} \\frac{A}{2}$ 的值为 $... | 467 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在凸四边形 $A B C D$ 中, $\\overrightarrow{B C}=2 \\overrightarrow{A D}$. 点 $P$ 是四边形 $A B C D$ 所在平面上一点, 满足\n$$\n\\overrightarrow{P A... | 2358 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{n+1}=(-1)^{n} n-a_{n}, n=1,2,3, \\cdots$, 且 $a_{10}=a_{1}$, 则 $a_{n} a_{n+1}$ 的最大值是 $\\qq... | 37 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $A B=6, B C=4$, 边 $A C$ 上的中线长为 $\\sqrt{10}$, 则 $\\sin ^{6} \\frac{A}{2}+\\cos ^{6} \\frac{A}{2}$的值为 $\... | 467 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在凸四边形 $A B C D$ 中, $\\overrightarrow{B C}=2 \\overrightarrow{A D}$. 点 $P$ 是四边形所在平面上一点, 满足 $\\overrightarrow{P A}+2020 \\overri... | 2358 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将正九边形的每个顶点等概率地涂上红、蓝两种颜色之一, 则存在三个同色的顶点构成锐角三角形的概率为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put you... | 503 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n正实数 $u, v, w$ 均不等于 1, 若 $\\log _{u}(v w)+\\log _{v} w=5, \\log _{v} u+\\log _{w} v=3$, 则 $\\log _{w} u$ 的值为 $\\qquad$.原始的答案是\\... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中, 点集 $K=\\{(x, y) \\mid x, y=-1,0,1\\}$. 在 $K$ 中随机取出三个点, 则这三个点两两之间的距离不超过 2 的概率为 $\\qquad$.原始的答案是\\frac{m}{n}... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设抛物线 $C: y^{2}=2 x$ 的准线与 $x$ 轴交于点 $A$, 过点 $B(-1,0)$ 作一直线 $l$ 与抛物线 $C$相切于点 $K$, 过点 $A$ 作 $l$ 的平行线, 与抛物线 $C$ 交于点 $M, N$, 则 $\\tr... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $\\sin ^{2} A+\\sin ^{2} C=2018 \\sin ^{2} B$, 则 $\\frac{(\\tan A+\\tan C) \\tan ^{2} B}{\\tan A+\\tan... | 2019 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $f(x)$ 是定义在 $(0,+\\infty)$ 上的单调函数, 对任意 $x>0$ 有 $f(x)>-\\frac{4}{x}, f\\left(f(x)+\\frac{4}{x}\\right)=3$,则 $f(8)=$ $\\qquad$... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n四面体 $A B C D$ 中, $A B \\perp B C, C D \\perp B C, B C=2$, 且异面直线 $A B$ 与 $C D$ 所成的角为 $60^{\\circ}$ 。若四面体 $A B C D$ 的外接球半径为 $\\s... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n过抛物线 $E: y^{2}=2 x$ 的焦点 $F$ 作两条斜率之积为 $-\\frac{1}{2}$ 的直线 $l_{1} 、 l_{2}$, 其中, $l_{1}$ 与抛物线 $E$ 交于 $A 、 C$ 两点, $l_{2}$ 与抛物线 $E$... | 16 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n棱长为 $\\sqrt{2}$ 的正四面体 $A B C D$ 中, 已知 $\\overrightarrow{A P}=\\frac{1}{2} \\overrightarrow{A B}, \\overrightarrow{A Q}=\\frac{... | 37 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知点 $P(x, y)$ 满足 $|x|+|y| \\leqslant 2$. 则到 $x$ 轴的距离 $d \\leqslant 1$ 的点 $P$ 的概率为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正三棱锥 $P-A B C$ 的高为 $P O, M$ 为 $P O$ 的中点, 过 $A M$ 作与棱 $B C$平行的平面，将三棱锥截为上、下两部分，试求此两部分的体积比.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终... | 25 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$\\cos ^{2} 10^{\\circ}+\\cos ^{2} 50^{\\circ}-\\sin 40^{\\circ} \\sin 80^{\\circ}=\\qquad$.The answer is in the form \\frac{m... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, 已知三个角 $A, B, C$ 成等差数列, 假设它们所对的边分别为 $a, b, c$, 并且 $c-a$ 等于 $A C$ 边上的高 $h$, 则 $\\sin \\frac{C-A}{2}=\\qq... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n实数 $x 、 y$ 满足 $4 x^{2}-5 x y+4 y^{2}=5$ ，设 $S=x^{2}+y^{2}$ ，则 $\\frac{1}{S_{\\max }}+\\frac{1}{S_{\\min }}$ 的值为$\\qquad$.原始的答案... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定曲线族 $2(2 \\sin \\theta-\\cos \\theta+3) x^{2}-(8 \\sin \\theta+\\cos \\theta+1) y=0, \\theta$ 为参数，求该曲线在直线 $y=2 x$ 上所截得的弦长的最大... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设数列 $\\left\\{a_{n}\\right\\}$ 是等差数列，数列 $\\left\\{b_{n}\\right\\}$ 满足 $b_{n}=a_{n+1} a_{n+2}-a_{n}^{2}, n=1,2, \\cdots$ 。\n（2）... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate\n$$\n\\lim _{n \\rightarrow \\infty} \\int_{0}^{1} \\int_{0}^{1} \\cdots \\int_{0}^{1} \\cos ^{2}\\left\\{\\frac{\\pi... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $C$ be a fixed unit circle in the Cartesian plane. For any convex polygon $P$ each of whose sides is tangent to $C$. let $... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each continuous function $f:[0,1]\\to\\mathbb{R}$, let $I(f)=$ $\\int _{0}^{1}x^{2}f( x)$ $dx$ and $J( x) = \\int _0^{1}x\... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n两人轮流投掷股子，每人每次投掷两颗，第一个使两颗股子点数和大于 6 者为胜，否则轮由另一人投郑。先投郑人的获胜概率是原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your... | 29 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 $f(x)=a x^{3}+b x^{2}+c x+d(a \\neq 0)$, 当 $0 \\leqslant x \\leqslant 1$ 时, $\\left|f^{\\prime}(x)\\right| \\leqslant 1$,... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $\\triangle A B C$ 的内角 $\\angle A 、 \\angle B 、 \\angle C$ 的对边分别为 $a 、 b 、 c$, 且 $\\angle A-\\angle C=\\frac{\\pi}{2}, a 、 b... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n从一个正方体的八个顶点中取出三个.则以这三个点为顶点构成直角三角形的概率是$\\underline{\\quad}$原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在一个圆上随机取三点, 则以此三点为顶点的三角形是锐角三角形的概率为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer on it... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在正三棱锥 $P-A B C$ 中, 已知侧棱与底面所成的角为 $45^{\\circ}$ 。则相邻两侧面所成角的余弦值为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRememb... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a 、 b 、 c \\in(0,1], \\lambda$ 为实数,使得\n$$\n\\frac{\\sqrt{3}}{\\sqrt{a+b+c}} \\geqslant 1+\\lambda(1-a)(1-b)(1-c).\n$$\n求入的最... | 91 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $f(x)$ 是定义在 $\\mathbf{R}$ 上的函数, 且对任意实数 $x$, 均有 $2 f(x)+f\\left(x^{2}-1\\right)=1$, 则 $f(\\sqrt{ 2})$ 的值为 $\\qquad$。\n#*原始的答... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $x, y, z$ 满足: $|x+2 y+3 z|=1$ 。\n求 $|x|^{2}+|y|^{2}+|z|^{2}+\\left|x^{2}+y^{2}+z^{2}\\right|$ 的最小值.\n#*原始的答案是\\frac{m}{n}的... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n袋中装有 2 个红球、 3 个白球、 4 个黄球，从中任取 4 个球，则其中三种颜色的球都有的概率是 $\\qquad$。原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put y... | 11 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在$\\triangle ABC$中，若$\\tan\\dfrac{A}{2}+\\tan\\dfrac{B}{2}=1$，则$\\tan\\dfrac{C}{2}$的最小值为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设$\\triangle ABC$的内角$\\angle A、\\angle B、\\angle C$的对边分别为$a、b、c$,且$\\angle A-\\angle C=\\frac{\\pi}{2},a、b、c$ 成等差数列.则$\\cos B$... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the greatest possible value of $\\sum_{i=1}^{10}\\cos(3x_i)$ for real numbers $x_1,x_2,\\ldots,x_{10}$ satisfying $\... | 529 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求所有的实数 $k$, 使得不等式 \n$$ \na^{3}+b^{3}+c^{3}+d^{3}+1 \\geqslant k(a+b+c+d) \n$$ \n对任意 $a 、 b 、 c 、 d \\in[-1,+\\infty)$ 都成立。... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n实数 $a 、 b$ 使得方程 \n$x^{3}-a x^{2}+b x-a=0\\quad①$ \n有三个正实根. 求 $\\frac{2 a^{3}-3 a b+3 a}{b+1}$ 的最小值.原始的答案是k\\sqrt{m}+n的形式，请给出k ... | 12 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知向量 $\\boldsymbol{a}=(x, 1), \\boldsymbol{b}=(2, y), \\boldsymbol{c}=$ $(1,1)$ ，且 $\\boldsymbol{a}-\\boldsymbol{b}$ 与 $\\bold... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中, $Rt \\triangle A B C$ 的三个顶点均在椭圆 $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$ $(a>b>0)$ 上, 其中, $A(0, b)$ 为... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在梯形 $A B C D$ 中, $B C / / A D, A B=A D$, $\\angle A B C=\\frac{2 \\pi}{3}, \\angle B C D=\\frac{\\pi}{2}$. 将 $\\triangle A B D... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n使得 $\\frac{x+3}{x^{2}-2 x+2}$ 为正整数的所有实数 $x$的和为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your ... | 133 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有五张分别写有 $1 、 2 、 3 、 4 、 5$ 数字的卡片，每次从中抽取一张，记下卡片上的数字，然后放回。这样抽取了四次，则抽到的最大数与最小数的差小于 4 的概率为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是... | 1056 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n利用24小时制计时，可得到某一时刻的四个数字之和比时刻$20:21$的四个数字之和小的概率为_______。原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your ans... | 301 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数$f(x)$的定义域为$\\mathbf{R}$，且$f(x-1)$为偶函数，$f(x-2)$为奇函数。当$x\\in[1,2]$时，$f(x)=ax^{2}+bx$。若$f(0)+f(3)=-1$，则$f\\left(\\frac{9}{2}\... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知复数 $z$ 、w 满足:\n$z+\\frac{20 \\mathrm{i}}{w}=5+\\mathrm{i}, w+\\frac{12 \\mathrm{i}}{z}=-4+10 \\mathrm{i}$ 。\n则 $|z w|$ 的最小值是... | 12 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知边长为 $a$ 的正方形 $A B C D$ 的顶点 $A$ 在平面 $\\beta$ 内，其余顶点均在平面 $\\beta$ 的同侧，且 $B 、 D$ 两点到平面 $\\beta$ 的距离分别为 $1 、 2$ 。若平面 $A B C D$ 与... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知复数 $z_{1}, z_{2}$ 满足:\n$$\n\\left|\\overline{z_{1}}\\right|=3,\\left|\\overline{z_{2}}\\right|=4,\\left|z_{1}+z_{2}\\right|=... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n从 $1 \\sim 9$ 这九个正整数中任选三个,则这三个数两两互素的概率是 $\\qquad$。原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer ... | 121 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设点 $A(2,0)$, 点 $B$ 在曲线 $y=\\sqrt{1-x^{2}}$上, $\\triangle A B C$ 是以 $A$ 为直角顶点的等腰直角三角形. 则 $|O C|$ 的最大值为 $\\qquad$.原始的答案是k\\sqrt{... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n从集合$\\{1,2,\\cdots,105\\}$中任取一个元素$a$，使得$x^{2}+ax+6a=0$只有整数解的概率为_______。原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemembe... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一只小虫在正八面体的表面上爬行，每秒从某一个顶点等可能地爬往四个相邻顶点之一。则小虫在第8秒爬回初始位置的概率为_______。原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to pu... | 299 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在正四面体 $P-A B C$ 中, 点 $E 、 F$ 分别在棱 $P B 、 P C$ 上. 若 $P E \\neq P F$ ，且 $A E=A F=2$ ， $E F=\\sqrt{3}$, 则四面体 $P-A E F$ 的体积为 _____... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate$$\\lim_{n\\rightarrow\\infty}\\int_{0}^{1}\\int_{0}^{1}\\cdots\\int_{0}^{1}\\cos^{2}\\left\\{\\frac{\\pi}{2n}\\left(x... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 $\\left\\{\\left.\\frac{5}{x}+y \\right\\rvert\\, 1 \\leqslant x \\leqslant y \\leqslant 5\\right\\}$ 中的最大元素与最小元素分别为 $M, N... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n双曲线 $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{16}=1$ 的左、右焦点分别为 $A, B, P$ 是双曲线上一点, 若 $\\triangle P A B$ 的内切圆圆心为 $(3,1)$, 则 $\\triangl... | 77 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将正九边形的每个顶点等概率地涂上红、蓝两种颜色之一, 则存在三个同色的顶点构成锐角三角形的概率为 $\\qquad$.\n原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put y... | 503 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\n在三角形 $A B C$ 中, $B C=4, C A=5, A B=6$, 则 $\\sin ^{6} \\frac{A}{2}+\\cos ^{6} \\frac{A}{2}=$ $\\qquad$. 原始的答案是\\frac{m}{n}的形... | 107 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若平面上非零向量 $\\vec{\\alpha}, \\vec{\\beta}, \\vec{\\gamma}$ 满足 $\\vec{\\alpha} \\perp \\vec{\\beta}, \\vec{\\beta} \\cdot \\vec{\... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个不均匀的骰子，掷出 $1,2,3,4,5,6$ 点的概率依次成等差数列.独立地先后掷该骰子两次，所得的点数分别记为 $a, b$ .若事件 \" $a+b=7$ \" 发生的概率为 $\\frac{1}{7}$ ，则事件 \" $a=b$ \"发生... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中, 设一条动直线 $l$ 与抛物线 $\\Gamma: y^{2}=4 x$ 相切，且与双曲线 $\\Omega: x^{2}-y^{2}=1$ 交于左、右两支各一点 $A 、 B$ 。求 $\\triangle A... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对有限非空实数集 $X$, 记 $X$ 的元素个数为 $|X|$ ，均有 \n$f(X)=\\frac{1}{|X|} \\sum\\limits_{a \\in X} a$. \n集合对 $(A, B)$ 满足 \n$A \\cup B=\\{... | 821 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $f(x)$ 是定义在 $(0,+\\infty)$ 上的单调函数, 对任意 $x>0$ 有 $f(x)>-\\frac{4}{x}, f\\left(f(x)+\\frac{4}{x}\\right)=3$,则 $f(8)= ?$ 原始的答案是\... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $f(x)$ 为定义在区间 $(0,+\\infty)$ 上的单调函数. 对于任意的 $x>0$ ，均有\n$f(x)>-\\frac{4}{x}, f\\left(f(x)+\\frac{4}{x}\\right)=3$.\n则 $f(8)= ?... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对于四面体 $A B C D, A B \\perp B C, C D \\perp$ $B C, B C=2$ ，且异面直线 $A B$ 与 $C D$ 所成的角为 $60^{\\circ}$. 若四面体 $A B C D$ 的外接球半径为 $\\s... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n过抛物线 $E: y^{2}=2 x$ 的焦点 F 作两条斜率之积为 $-\\frac{1}{2}$ 的直线 $l_{1}, l_{2}$, 其中 $l_{1}$ 交 $E$ 于 $A, C$ 两点, $l_{2}$ 交 E 于 $B, D$ 两点, ... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有一人进行投篮训练，投篮 5 次，失误一次扣 1 分，进一次得 1 分，连进 2次得 3 分，连进 3 次得 5 分. 若投篮的命中率为 $\\frac{2}{5}$ ，则投篮 3 次恰好得 2 分的概\n率为 .原始的答案是\\f... | 149 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数， $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\\max ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $PQR$ be an equilateral triangle with sides of length three units. $U$, $V$, $W$, $X$, $Y$, and $Z$ divide the sides into ... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\n已知实数 $x_{1}, x_{2}, \\cdots, x_{2018}$ 满足：对任意的 $1 \\leqslant i<j \\leqslant 2018$, 均有 $x_{i}+x_{j} \\geqslant(-1)^{i+j}$. 求 ... | 2037173 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $x 、 y 、 z \\in(0,1)$ ，满足 $\\sqrt{\\frac{1-x}{y z}}+\\sqrt{\\frac{1-y}{z x}}+\\sqrt{\\frac{1-z}{x y}}=2$. 求 $x y z$ 的最大值. ... | 91 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对区间 $[0,1]$ 上的实数 $x 、 y 、 z$, 记 $M$ 为 $x-x y-x z+y z 、 y-y x-y z+x z 、 z-z x-z y+x y$中的最小数. 求 $M$ 的最大可能值. 原始的答案是\\frac{m}{n}的... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正数 $a 、 b 、 c$ 满足 $a^{2}+b^{2}+2 c^{2}=1$ .则 $\\sqrt{2} a b+2 b c+7 a c$ 的最大值为 $\\qquad$. 原始的答案是k\\sqrt{m}+n的形式，请给出k + m + ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个盒中有 9 个正品和 3 个废品, 每次取一个产品, 取出后不再放回。则在取出 2个正品前已取出 3 个废品的概率为 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemembe... | 56 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在直三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=1, B C=C C_{1}=\\sqrt{3}, \\angle A B C=90^{\\circ}$, 点 $P$ 是平面 $A B C$上一动点, 则 $A_{1} ... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $x, y, z$ 满足: $|x+2 y+3 z|=1$ 。 \n求 $|x|^{2}+|y|^{2}+|z|^{2}+\\left|x^{2}+y^{2}+z^{2}\\right|$ 的最小值. 原始的答案是\\frac{m}{n}的... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的正实数 $\\alpha$, 对于满足 $a+b+c \\mid a^{2}+b^{2}+c^{2}$ 的不全相等的正整数 $a, b, c$, 都有方程 \n$$ \n(x+b)^{2}=(x+c)(x+a) \n$$ \n在区间 $... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将编号为 $1,2,3, \\cdots, 9$ 的九个小球随机放置在圆周的九个等分点上，每个等分点上各放一个小球，设圆周上所有相邻两个球号码之差的绝对值之和为 $S$ ，求使 $S$ 达到最小值的放法的概率。（注：如果某种放法，经旋转或镜面反射后与另... | 316 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若函数 $f(x)=\\frac{x}{\\sqrt{1+x^{2}}}$ 且 $f^{(n)}(x)=\\underbrace{f[f[f \\cdots f(x)]]}_{n}$, 则 $f^{(99)}(1)=$ $\\qquad$.原始的答案是... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某情报站有 $A, B, C, D$ 四种互不相同的密码, 每周使用其中的一种密码, 且每周都是从上周未使用的三种密码中等可能地随机选用一种。设第1周使用 $A$ 种密码, 那么第 7 周也使用 $A$ 种密码的概率是 $\\qquad$. (用最简分... | 304 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某情报站有 $A, B, C, D$ 四种互不相同的密码，每周使用其中的一种密码，且每周都是从上周未使用的三种密码中等可能地随机选用一种。设第1周使用 $A$种密码，那么第7周也使用 $A$ 种密码的概率是 $\\qquad$ (用最简分数表示)。原始... | 304 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n从 $1,2, \\cdots, 20$ 中任取 5 个不同的数, 其中至少有两个是相邻数的概率为原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer o... | 555 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若从 $1,2,3,4,5,6,7,8,9$ 中随机地取出四个两两互异的数，分别记为 $a, b, c, d$ ，则 $a+b$ 与 $c+d$ 奇偶性相同的概率是 $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个半径为 1 的小球在一个内壁棱长为 $4 \\sqrt{6}$ 的正四面体容器内可向各个方向自由运动,则该小球永远不可能接触到的容器内壁的面积是 $\\qquad$.原始的答案是k\\sqrt{m}+n的形式，请给出k + m + n的值。\n\n... | 75 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $E, F$ 分别是 $A C, A B$ 的中点, $A B=\\frac{2}{3} A C$. 若 $\\frac{B E}{C F}<t$ 恒成立, 则 $t$ 的最小值为 $\\qquad$.\... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设等差数列 $\\left\\{a_{n}\\right\\}$ 的前 $n$ 项和为 $S_{n}$, 若 $S_{2010}-S_{1}=1$, 则 $S_{2011}=$ $\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n... | 4020 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $A, B, C, D$ 是空间四个不共面的点, 以 $\\frac{1}{2}$ 的概率在每对点之前连一条边, 任意两对点之间是否连边是相互独立的, 则 $A, B$ 可用空间折线（一条或若干条边组成的）连接的概率为 $\\qquad$。\n原始... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{1}=2, a_{n+1}=\\frac{2(n+2)}{n+1} a_{n}\\left(n \\in \\mathbb{N}^{*}\\right)$, 则 $\\frac{... | 4028 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{1}=2, a_{n+1}=\\frac{2(n+2)}{n+1} a_{n}\\left(n \\in \\mathbf{N}^{*}\\right)$ ，则 $\\frac{... | 4028 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正四棱雉 $S-A B C D$ 的侧棱长为 $4, \\angle A S B=30^{\\circ}$, 在侧棱 $S B 、 S C 、 S D$ 分别取点 $E 、 F 、 G$. 则空间四边形 $A E F G$ 周长的最小值为 $\\q... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nToday, Ivan the Confessor prefers continuous functions $f:[0,1] \\rightarrow \\mathbb{R}$ satisfying $f(x)+f(y) \\geq|x-y|$ fo... | 5 |
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